Geometry and Topology Seminar

Traditionally, the Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
In Spring of 2016, the seminar will meet on Thursdays at 2:25PM in B 231 of Van Vleck.
For more information, contact Tullia Dymarz or Alexandra Kjuchukova.

Spring Abstracts

Ryan Blair

Distance and Exceptional Surgeries on Knots

Distance is a measure of complexity for a bicompressible surface in a 3-manifold which is defined using the curve complex for the surface. Recently, distance has been used to better understand Dehn surgery on knots in 3-manifolds. In particular, I will present results which show that knots with high distance surfaces do not admit non-hyperbolic surgeries or cosmetic surgeries. Applications to the cabling conjecture and the Berge conjecture will also be discussed.

Fall Abstracts

Hung Tran

Relative divergence, subgroup distortion, and geodesic divergence

In my presentation, I introduce three new invariants for pairs $(G;H)$ consisting of a finitely generated group $G$ and a subgroup $H$. The first invariant is the upper relative divergence which generalizes Gersten's notion of divergence. The second invariant is the lower relative divergence which generalizes a definition of Cooper-Mihalik. The third invariant is the lower subgroup distortion which parallels the standard notion
of subgroup distortion. We examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We also explore relative divergence of $CAT(0)$ groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups. We answer the question of Behrstock and Drutu about the existence of Morse geodesics in $CAT(0)$ spaces with divergence function strictly greater than $r^n$ and strictly less than $r^{n+1}$, where $n$ is an integer greater than $1$. More precisely, we show that for each real number $s>2$, there is a $CAT(0)$ space $X$ with a proper and cocompact action of some finitely generated group such that $X$ contains a Morse bi-infinite geodesic with the divergence equivalent to $r^s$.

Tullia Dymarz

Non-rectifiable Delone sets in amenable groups

In 1998 Burago-Kleiner and McMullen constructed the first
examples of coarsely dense and uniformly discrete subsets of R^n that are
not biLipschitz equivalent to the standard lattice Z^n. Similarly we
find subsets inside the three dimensional solvable Lie group SOL that are
not bilipschitz to any lattice in SOL. The techniques involve combining
ideas from Burago-Kleiner with quasi-isometric rigidity results from
geometric group theory.

Jesse Wolfson

Counting Problems and Homological Stability

In 1969, Arnold showed that the i^{th} homology of the space of un-ordered configurations of n points in the plane becomes independent of n for n>>i. A decade later, Segal extended Arnold's method to show that the i^{th} homology of the space of degree n holomorphic maps from \mathbb{P}^1 to itself also becomes independent of n for large n, and, moreover, that both sequences of spaces have the same limiting homology. We explain how, using Weil's number field/function field dictionary, one might have predicted this topological coincidence from easily verifiable statements about specific counting problems. We then discuss ongoing joint work with Benson Farb and Melanie Wood in which we use other counting problems to predict and discover new instances of homological stability in the topology of complex manifolds.

Matthew Cordes

Morse boundaries of geodesic metric spaces

I will introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with equivalence classes of geodesic rays that identify the ``hyperbolic directions" in that space. (A ray is Morse if quasi-geodesics with endpoints on the ray stay bounded distance from the ray.) This boundary is a quasi-isometry invariant and a visibility space. In the case of a proper CAT(0) space the Morse boundary generalizes the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. Time permitting I will also discuss some results on Morse boundary of the mapping class group and briefly describe joint work with David Hume developing a capacity dimension for the Morse boundary.

Anton Izosimov

Stability for the multidimensional rigid body and singular curves

A classical result of Euler says that the rotation of a
torque-free 3-dimensional rigid body about the short or the long axis is
stable, while the rotation about the middle axis is unstable. I will
present a multidimensional generalization of this result and explain how
it can be proved using some basic algebraic geometry of singular curves.

Jacob Bernstein

Hypersurfaces of low entropy

The entropy is a quantity introduced by Colding and Minicozzi and may be thought of as a rough measure of the geometric complexity of a hypersurface of Euclidean space. It is closely related to the mean curvature flow. On the one hand, the entropy controls the dynamics of the flow. On the other hand, the mean curvature flow may be used to study the entropy. In this talk I will survey some recent results with Lu Wang that show that hypersurfaces of low entropy really are simple.

Yun Su

Higher-order degrees of hypersurface complements.

Gao Chen

Classification of gravitational instantons

A gravitational instanton is a noncompact complete hyperkahler manifold of real dimension 4 with faster than quadratic curvature decay. In this talk, I will discuss the recent work towards the classification of gravitational instantons. This is a joint work with X. X. Chen.

Dan Cristofaro-Gardiner

Higher-dimensional symplectic embeddings and the Fibonacci staircase

McDuff and Schlenk determined when a four dimensional symplectic ellipsoid can be embedded into a ball, and found that when the ellipsoid is close to round, the answer is given by an infinite staircase determined by the odd-index Fibonacci numbers. I will explain joint work with Richard Hind, showing that a generalization of this holds in all even dimensions.

Danny Ruberman

Configurations of embedded spheres

Configurations of lines in the plane have been studied since antiquity. In recent years, combinatorial methods have been used to decide if a specified incidence relation between certain objects ("lines") and other objects ("points") can be realized by actual points and lines in a projective plane over a field. For the real and complex fields, one can weaken the condition to look for topologically embedded lines (circles in the real case, spheres in the complex case) that meet according to a specified incidence relation. I will explain some joint work with Laura Starkston (Stanford) giving new topological restrictions on the realization of configurations of spheres in the complex projective plane.

Quinton Westerich

Harmonic Chern Forms on Polarized Kähler Manifolds

Abstract: The higher K-energies are functionals whose critical points
give Kähler metrics with harmonic Chern forms. In this talk, we relate
the higher K-energies to discriminants and use the theory of stable
pairs to obtain results on their boundedness and asymptotics.

Tommy Wong

Milnor Fiber of Complex Hyperplane Arrangement

The existence of Milnor fibration creates rooms and provides a platform to discuss the topology of complex algebraic varieties. In this talk, the study of hyperplane arrangements will be specified.
Many open questions have been raised subject to the Milnor fiber of the mentioned fibration. For instance,while the homology of the arrangement complement can be described by the Orlik-Soloman Algebra, which is combinatorically determined by the intersection poset, it has been conjectured that the poset also determines the homology of the Milnor fiber.
There are active work on this open conjecture, especially in C^3. Several classical results will be mentioned in the talk. A joint work with Su, serving as an improvement of some of the classical work, will also be briefly described.